No, the acceleration is the same, and so masses of any weight fall at the same speed after a given time. To see this consider two masses, $m$ and $M$, and assume that $M\gg m$ or, equivalently, that $M$ is fixed in space somehow (this is to avoid the problems associated with $M$ moving which causes heavier objects to effectively fall faster as discussed in the other question.
So, what is the (magnitude of the) force due to gravity between $m$ and $M$? Well, Newton talls us that it is
$$F = \frac{GmM}{r^2}$$
So, now, what is the (magnitude of the) acceleration of $m$? Well, Newton also tells us that $F = ma$, so $a = F/m$, so
$$\begin{align}
a &= \frac{GmM}{mr^2}\\
&= \frac{GM}{r^2}
\end{align}$$
And look: there is no dependence on $m$ at all, because it cancels out.
That this is true is actually a surprising fact: there are, really, two definitions of mass here:
- inertial mass is the mass which occurs in $F = ma$, Newton's second law;
- gravitational mass is the mass which occurs in $F = Gm_1 m_2/r^2$, Newton's law of gravitation.
And it turns out that these two masses are always the same for a given object (or, strictly, are always in the same proportion, with the constant of proportionality being absorbed into $G$). The above result, that all things fall with the same acceleration, is because of this equality between inertial and gravitational mass.
This is not something that needs to be the case: experimentally it is the case to the best of our knowledge (ie no experiments we have been able to do have ever shown it not to be the case), but nothing says it has to be the case. If it turned out not the case then General Relativity, for which this equality is a rather basic assumption, would fail.
(Just to be clear: although I've made the point that the equality of inertial and gravitational mass is an experimental fact, which I think is important to undertand, I strongly believe it to be true. I'm not trying to push some alternative theory of gravity!)
